SIT787:Mathematics for AI - Science And Maths Assignment Help

Assignment Task

Question 1) Consider these vectors:

1. Determine which two vectors are most similar to each other based on these norms:

2. Determine which two vectors are most similar to each other based on the cosine similarity measure:

[You can use decimals here. However, you need to show the final answer in terms of quotients and surds before converting them into decimals.]

3. Explain the reason behind the difference in result between (i) and (ii) if you observe any.
4. Can the difference be resolved? Give details of your suggestion, if you have any, and explain the outcome if your suggestions are applied.

Question 3) Consider the following matrix with a parameter c.

1. Based on the values of b and c, determine when the system

has a unique solution, no solutions, and many solutions.

Now fix c = 1, and answer the following questions (ii), (iii), and (iv).

2. Find orthonormal bases for the column space C (A), the row space C (A^T), the null space N(A) and the left null space N(AT ).

Hint: For a m × n matrix A, the null space N (A) is defined as {x ? R n |Ax = 0}. For the same matrix, the left nulls pace N (A^T) is defined as {y ? R m|A^T y = 0}.

3. Find the dimension of these vector spaces.

4. Show that any vector in C (A) is perpendicular to any vector in N (A^T).

Question 4) Consider the following matrix.

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